Diophantine Equations with a Finite Number of Solutions: Craig Smorynski's Theorem, Harvey Friedman's Conjecture and Minhyong Kim's Guess
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Matiyasevich's theorem states that there is no algorithm to decide whether or not a given Diophantine equation has a solution in non-negative integers. Smorynski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. We prove: (1) Smorynski's theorem easily follows from Matiyasevich's theorem, (2 ) Hilbert's Tenth Problem for Q has a negative solution if and only if the set of all Diophantine equations with a finite number of rational solutions is not recursively enumerable.
2019 ◽
2019 ◽
2019 ◽
2018 ◽
2018 ◽
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2015 ◽
Vol 92
(2)
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pp. 187-194
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